Factorization of Perfect Square Trinomials

In factorization of perfect square trinomials we will learn how to solve the algebraic expressions using the formulas. To factorize an algebraic expression expressible as a perfect square, we use the following identities:

(i) a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b)

(ii) a2 - 2ab + b2 = (a - b)2 = (a - b) (a - b)

Note: We will also learn to use two identities in the same question, to factorize the expression.

Solved problems on factorization of perfect square trinomials:

1. Factorization when the given expression is a perfect square:

(i) x4 - 10x2y2 + 25y4

Solution:

We can express the given expression x4 - 10x2y2 + 25y4 as a2 - 2ab + b2

= (x2)2 - 2 (x2) (5y2) + (5y2)2

Now it’s in the form of the formula of a2 + 2ab + b2 = (a + b)2 then we get,

= (x2 - 5y2)2

= (x2 – 5y2) (x2 – 5y2)

(ii) x2+ 6x + 9

Solution:

We can express the given expression x2 + 6x + 9 as a2 + 2ab + b2

= (x)2 + 2 (x) (3) + (3)2

Now we will apply the formula of a2 + 2ab + b2 = (a + b)2 then we get,

= (x + 3)2

= (x + 3) (x + 3)

(iii) x4 - 2x2 y2 + y4

Solution:

We can express the given expression x4 - 2x2 y2 + y4 as a2 - 2ab + b2

= (x2)2 - 2 (x2) (y2) + (y2)2

Now we will apply the formula of a2 - 2ab + b2 = (a - b)2 then we get,

=(x2 – y2)2

=(x2 - y2) (x2 – y2)

Now we will apply the formula of differences of two squares i.e. a2 - b2 = (a + b) (a – b) then we get,

= (x + y) (x- y) (x + y) (x- y)

2. Factorize using the identity:

(i) 25 – x2 - 2xy - y2

Solution:

25 – x2 - 2xy - y2

= 25 - [x2 + 2xy + y2], rearranged

Now we see that x2 + 2xy + y2 as in the form of a2 + 2ab + b2.

= (5)2 – (x + y)2

Now we will apply the formula of differences of two squares i.e. a2 - b2 = (a + b) (a – b) then we get,

= [5 + (x + y)] [5 - (x + y)]

= (5 + x + y) (5 – x - y)

(ii) 1- 2xy- (x2 + y2)

Solution:

1- 2xy- (x2 + y2)

= 1 - 2xy - x2 - y2

= 1 - (x2 + 2xy + y2), rearranged

= 1 - (x + y )2

= (1)2 – (x + y)2

= [1 + (x + y)] [1 - (x + y)]

= [1 + x + y] [1 - x - y]

Note:

We see that to solve the above problems on factorization of perfect square trinomials we not only used perfect square identities but we also used the difference of two squares identity in different situations.